Crack the eggs into a medium bowl. Add 3 tablespoons water, 1 teaspoon soy sauce, 1 teaspoon sesame oil, 1 teaspoon salt, and 1/4 teaspoon white pepper. Whisk until uniform and slightly frothy. Set aside.

Filling

Prepare shrimp and vegetables

If using whole shrimp, chop into roughly 1/4- to 1/2-inch pieces. Slice green onions, keeping whites and greens separate. Drain and finely chop water chestnuts and mushrooms.

Heat a large nonstick skillet or well-seasoned wok over medium-high heat. Add 1 tablespoon vegetable oil. When hot, add minced garlic and grated ginger; stir-fry 15–20 seconds until fragrant.

Add the chopped shrimp and the white parts of the green onions. Stir-fry 1–2 minutes until shrimp are mostly opaque (they may not be fully cooked through). Add mushrooms and water chestnuts and cook another 1 minute. Add bean sprouts and toss briefly just to warm and combine. Season lightly with 1/4 teaspoon salt and a pinch of pepper. Transfer the filling to a bowl and wipe out the pan if necessary.

Add remaining 2 tablespoons vegetable oil to the skillet and return to medium heat. Swirl to coat. Pour a thin layer (about 1/3 cup) of the egg mixture into the skillet to form a small omelet (roughly 6-inch diameter). Immediately spoon about 2–3 tablespoons of the shrimp-vegetable filling onto one half of the egg.

Cook until the egg is mostly set on the bottom and the top is still slightly wet, about 1–2 minutes. Fold the egg over the filling to form a half-moon. Press gently with a spatula and cook another 20–30 seconds. Transfer to a plate and keep warm. Repeat to make 3–4 egg foo yung patties, using 2–3 tablespoons filling per omelet and about 2 eggs' worth of mixture for each (adjust to yield 3–4 patties).

Sauce (for serving)

In a small bowl, whisk 1 teaspoon cornstarch with 1 tablespoon cold water until smooth and set aside.

In a small saucepan over medium heat, combine 1 cup chicken or vegetable stock, 2 tablespoons soy sauce, 1 tablespoon oyster sauce, 1 teaspoon rice vinegar, and 1 teaspoon sugar. Bring to a gentle simmer.

Whisk the cornstarch slurry again and slowly pour into the simmering sauce while whisking. Continue to simmer 1–2 minutes until the sauce thickens and becomes glossy. Stir in 1/2 teaspoon sesame oil. Taste and adjust seasoning (add a pinch of sugar or a splash more soy sauce if needed). Keep warm.

To Serve

Place 1–2 shrimp egg foo yung patties on each serving plate. Spoon the warm sauce over the patties or serve the sauce on the side for dipping.

Garnish with sliced green onion greens and chopped cilantro or parsley. Serve immediately with steamed rice.